IEEE Robotics & Automation Magazine - June 2012 - 27
Current
Potential Field
νw, 1
νθ
νw, 4
ICR
Yb
νy
yICR
Xb
xICR
Ob
Possible
Wheel Location
νx
νw, 2
νw, 3
Figure 8. The consistent motion of the mobile platform around
the ICR is depicted in the platform body frame (superscript b).
The whole structure is one rigid body since the leg lengths are
kept constant here.
demonstration is provided in Figure 13, wherein the wheel
center point moves from A to B. In this example, the deviations from the nominal trajectory are 32% below the values
of locked legs. A more detailed discussion on all these
experimental results can be found in [23].
Task Execution
As a benefit of the autonomous whole-body coordination
of our highly redundant robot, a task can be defined in the
low-dimensional Cartesian space of the TCPs. We implemented a passive impedance control [32] that complies
with the following law:
simp
@Vimp (q, X odo , t) T
_
¼ g(q) À
ÀDimp (q)q:
@q
Admissible Potential
Field Extension
Figure 10. Feasible potential field extensions for the ICR
repulsion. The extension depends on the location of the wheels
to prevent obstructive overlaps [23].
mobility of the manipulator, the odometry X odo is required
here. Additional damping is injected through Dimp (q). From
a passivity point of view, any positive semidefinite matrix can
be applied. Within this work, Dimp (q) is chosen such that the
desired damping ratios are realized. That is again achieved by
utilizing the double diagonalization [29] of the effective mass
matrix and the Hessian of the potential function, i.e.,
@ 2 Vimp (q, X odo , t)=@q2 , in task coordinates.
The potential according to the spatial spring that is
spanned between two frames H 1 2 SE(3) and H 2 2 SE(3)
is denoted by Vs (H 1 , H 2 , K), wherein K represents the
(5)
∗ ∗
Gravity effects are compensated by g(q). A spatial spring
that is spanned between the actual and desired TCP frames is
defined by the potential Vimp (q, X odo , t). Owing to the
0.4
∗
tstart
0.8
+
t = 15 s
+
Wheel 4
Potential
y (m)
tend
-0.4
∗
Wheel 1
+
0
t=9s
+
Wheel 3
t = 18 s
+
Wheel 2
-0.8
-0.8
Xb
0.4
y (m)
∗∗
Yb
0.8
0
-0.4
-0.8
-0.8
-0.4
0
x (m)
0.4
0.8
Figure 9. Repulsive potentials placed around the wheel center
points to repel the ICR from the wheels [23].
-0.4
0
x (m)
0.4
0.8
Hardware (ICR Control On)
Simulation (ICR Control On)
Simulation (ICR Control Off)
Figure 11. The ICR in the platform body frame. Activated ICR
control avoids the singular locations around the wheel center
points.
JUNE 2012
*
IEEE ROBOTICS & AUTOMATION MAGAZINE
*
27
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